displacement detail problem
I have a few possibly helpful pointers:


Foremost, I have noticed that Blender doesn't always play nicely with .png files (especially with normals and displacements). Try a .tiff, or other type.
I had the same issue of 'undulating surfaces', surfaces that should be smooth, but appear bumpy for no apparent reason.
Second, I notice thin, horizontal rectangles in your second image. Check that your mesh geometry is composed of squares, not rectangles. Otherwise you'll experience issues when displacing or otherwise deforming the mesh.
Lastly, double check that your UV's are unwrapped properly, same reason as above. If the UV is stretched in one direction and not another, it will distort the effect. Not sure if this is the problem here but thought I'd suggest in case.


One more thing: you might want to try using adaptive subdivision in this case.
Andrew Price does a good job explaining what it does and how to use it in this video.

I hope one of these helps!